Prozorov_23 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 23 Power and resonance in RLC circuit power in a resistor (review) v2 t 2 the instantaneous power is p t v t i t Ri t R T energy dissipated in one cycle (over period T) is: U p t dt , T 0 with harmonic variation, 1 2 f i t I cos t Average power (over the cycle): 1 P p t dt I 2 R T 0 2 T 1 2 I R 2 VRMS 2 cos 2 t dt 0 2 2 VRMS I 2R 2 cos d 2 RI RMS R 0 2 V I , I RMS 2 2 PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 2 exercise: LRC circuit – find R, L, C VR RI R 3 Ohm VC X C I X C 2 Ohm VL X L I X L 6 Ohm 1 1 XC C 50 mF C X C X L L L XL 0.6 mH 10 rad / s, I 10 A 4 PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 3 Power in an inductor PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 4 example: RL circuit An RL circuit is driven by an AC generator as shown in the figure. The current through the resistor and the generator voltage are: A. always out of phase B. always in phase C. sometimes in phase and sometimes out of phase E VL And this is the current through all elements. PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University I VR 17 October 2011 5 low- and high-pass filters Vout Vout depends on frequency: C High smaller reactance VC = Vout 0 Low larger reactance no current flows through R smaller VR VC = Vout Vout R This is a circuit that only passes low frequencies: low-pass filter Bass knob on radio 0 1 RC If instead we look at the voltage through the resistor: high-pass filter Treble control (We can also use RL circuits are high- and low-pass filters) PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 6 a - filter 1 XC C X L L if you want to filter any frequencies, but let alone the DC component: • The capacitor C1 offers low reactance to the AC component of the input while it offers infinite reactance to the DC component. As a result the capacitor shunts an appreciable amount of the AC component while the DC component continues its journey to the inductor L. • The inductor L offers high reactance to the AC component but it offers almost zero reactance to the DC component. As a result the DC component flows through the inductor while the AC component is blocked. • The capacitor C2 bypasses the AC component which the inductor had failed to block. As a result only the DC component appears across the load RL. PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 7 the resonance L Current amplitude in a series RLC circuit driven by a source of amplitude E : I R C E Z Maximum current when impedance Z XL XC R 2 is minimum 2 i.e., when XL XC L 0 1 C 1 LC Resonance: Driving frequency = natural frequency PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 8 circuit behavior at resonance PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 9 tuning a radio PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 10 band pass filter I E E cos Z R Maximum current E R I 0 maximum cos (cos ~ 1) 0 0 1 2o ~ 0 (circuit in phase) LC Resonance Low I ~ 0 due to capacitor High I ~ 0 due to inductor PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 1 LC 17 October 2011 11 power in AC circuits Instantaneous power supplied to the circuit: P t t i t E cos t I cos t Often more useful: Average power 1 P EI cos 2 P t EI cos t cos t cos t cos t cos t cos sin t sin cos t cos2 x 1 2 cos x sin x Define: E rms E 2 Irms 2 0 1 2 cos2 xdx 2 0 1 2 cos x sin xdx 0 I 2 PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University P Erms Irms cos 17 October 2011 12 power factor P Erms Irms cos Power factor (PF) Maximum power = 0 I Resonant circuit E E cos Z R P Irms R 2 All energy dissipation happens at the resistor(s). PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 13 the “quality” Q factor How “sharp” is the resonance? (ie, the resonance peak) Umax Q 2 U • For RLC circuit, U max Umax is max energy stored in the system ΔU is the energy dissipated in one cycle 1 LI 2 2 period 1 2 1 2 2 • Losses only come from R : U I RT I R 2 2 0 0 1 LC 0L XL,0 Q R R XC ,0 Q 0CR R 1 PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University Q XL,0 XC ,0 R R 17 October 2011 14 Q XL,0 R XC ,0 R Large Q sharp peak better “quality” L and C control how much energy is stored. R controls how much energy is lost. Small resistance Large Q PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 15 transformers Application of Faraday’s Law • Changing EMF in primary coil creates changing flux • Changing flux creates EMF in secondary coil. V N Magnetic flux remains mostly in the core. Core “directs” B lines d B for both coils dt d B V1 V2 dt N1 N2 V1 V2 N1 N2 V1 Efficient method to change voltage for AC. N1 N2 V2 d B dt If no energy is lost in the coils, power on both sides must be the same V1I1 V2I2 PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 16 transformers PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 17 Transformers PHYS222 - Lecture 23 - Prof. Ruslan Prozorov - Iowa State University 17 October 2011 18 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.

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